Video Transcript
In this video, we will learn how to
describe the reflection of light rays from a convex mirror. First, let’s see how a convex
mirror looks.
Seen from the side, such a mirror
could look this way. Were we to shine a ray of light on
the mirror, it would reflect or bounce off. Let’s imagine that instead of
looking at this mirror from the side, we look at it from above like this. From that perspective, here’s how
the mirror would appear. This mirror is called convex
because the middle of the mirror, here, is closer to an observer than the edges of
the mirror are. We could say that the middle of the
mirror bends toward the observer. This is a convex mirror.
At the same time, this is a
spherical mirror. We can say that because the surface
of the mirror, here, is a small part of a sphere. The mirror’s full name is a convex
spherical mirror. This point here is the center of
the sphere that our mirror helps to make. It has a special name, the center
of curvature. The distance between the center of
curvature in any point on the surface of the mirror is the same. That is, this distance here equals
this distance here, equals this distance here, equals this distance here, and so
on. The name for the distance between
the center of curvature and a point on the mirror surface is the radius of
curvature.
Now, we know that the job of a
mirror is to reflect light. Say we have some incoming light
rays that are parallel to one another. When these rays reach the mirror,
they’ll bounce off. This top ray of light reaches the
mirror at the center of the mirror surface. What will happen is that this ray
reflects directly backward along the path that it traveled to reach the mirror. The next parallel ray of light,
once it reaches the mirror, will reflect off its surface like this. And then the last ray will bounce
off the mirror in this direction. The three reflected rays never
meet. In fact, they get farther and
farther apart as they travel. Because the reflected rays don’t
meet, they don’t form an image.
So then, what is the image we see
when we look at an object’s reflection in a mirror? Say that we take our three
reflected rays, and we trace them backward. These dashed lines do meet at a
point. Since this is where the reflected
rays would meet if they could travel back into the mirror, we call this the focal
point. Whenever we look at a mirror and
see an image, that image seems to be at the mirror’s focal point.
So far, we’ve identified three
important points on this diagram: the center of curvature, the focal point, and this
point here, which is the center of the mirror surface. The distance between the center of
the surface of the mirror and the focal point is called the focal length. For any convex spherical mirror,
the focal length is exactly one-half the radius of curvature. Knowing all this about convex
spherical mirrors, let’s look now at a few examples.
Below is a ray diagram for a convex
mirror. Which one of the five locations
along the optical access represents the focal point of the spherical mirror?
Since this mirror is convex, that
means an observer would be on this side of the mirror. Since the mirror is spherical, that
means its surface is part of a larger sphere. In this example, we have five
locations, one, two, three, four, five, along what is called the optical axis. That axis follows this line that
goes right through the center of the mirror. We want to figure out which of
these five locations represents the focal point of the mirror. Notice that the diagram shows these
three parallel rays of light running into the mirror. The top ray reflects off in this
direction, the middle ray reflects straight back along the path it came, and the
bottom ray reflects off the mirror like this.
To find the location of the focal
point, we’ll trace these three reflected rays backward. The top ray is traced backward like
this, the middle ray like this, and the bottom ray this way. We’ve only traced back far enough
to find the point where the rays meet, where they cross. That point is here, and this is the
focal point of the mirror. We see this is at the location
marked four. Location four is the mirror’s focal
point.
Let’s look now at another
example.
The radius of curvature of a convex
mirror is five centimeters. Which one of the following
sentences about the focal length is correct? (A) The focal length is five
centimeters and is the distance from the center of the surface of the mirror to the
focal point. (B) The focal length is five
centimeters and is the distance from the center of the surface of the mirror to the
center of curvature. (C) The focal length is 2.5
centimeters and is the distance from the center of the surface of the mirror to the
center of curvature. (D) The focal length is 2.5
centimeters and is the distance from the center of the surface of the mirror to the
focal point.
Knowing that we have a convex
mirror with a radius of curvature of five centimeters, we can clear space at the top
of our screen and sketch this mirror. Let’s say that this is our convex
mirror, and here is the mirror center of curvature. The distance between this point and
the point at the center of the mirror here is called the radius of curvature. In this example, the radius of
curvature is given as five centimeters. Now, all of our answer options
describe the mirror’s focal length. This is different from the radius
of curvature.
If we sketch it on our diagram,
this mirror’s focal point would be a point here. It’s the point halfway between the
center of curvature and the center of the surface of the mirror. Then, the distance between the
center of the surface of the mirror and our focal point, that distance is called the
focal length of the mirror. As an equation, we can say that
focal length equals one-half times the radius of curvature. Since the radius of curvature of
this mirror is five centimeters, then the focal length must be half of that, or 2.5
centimeters.
Only two of our answer options, (C)
and (D), have the focal length at 2.5 centimeters. This means we can cross out options
(A) and (B). The difference between our two
remaining answer choices is that one says the focal length is the distance from the
center of the surface of the mirror to the center of curvature, while the other has
it as the distance from the center of the surface of the mirror to the focal
point. To see the difference between these
choices, let’s look at an up close view of our sketch.
Okay, in this view, we have our
center of curvature, the focal point of the mirror, and then here is the center of
the surface of the mirror. Answer option (C) says that the
focal length of this mirror is measured as this distance. We can see, though, that this can’t
be correct. That’s because this distance in
pink is the radius of curvature, five centimeters. The focal length is one-half that
distance. The true focal length is the
distance from the center of the surface of the mirror to the focal point. This is described by option
(D). The focal length is 2.5 centimeters
and is the distance from the center of the surface of the mirror to the focal
point.
Let’s look now at one last
example.
Which one of the following
sentences is the correct description for what happens to parallel rays incident on a
convex mirror? (A) They will continue
undisturbed. (B) They will be focused at a point
which is called the focal point. (C) They will not be focused at a
point, but the mirror will still have a focal point. (D) They will not be focused at a
point, and the mirror will have no focal point.
To see which answer is correct,
let’s clear some space at the top of our screen. And we can draw parallel rays
incident on a convex mirror. Because these rays are reaching a
mirror, they will reflect off of it. The center ray will bounce straight
backward in this direction, while the top and bottom rays will reflect like
this.
Looking again at our answer
choices, we see that answer (A) can’t be correct. These rays do not continue
undisturbed; instead, they reflect off the mirror. Option (B) says that the rays are
focused to a point, but we see that these reflected rays travel farther and farther
away from one another over time. Choice (B) can’t be correct
either. To figure out which of options (C)
and (D) is correct, we need to know whether this mirror has a focal point. To find out, let’s trace the
reflected rays backward using dashed lines.
Tracing the center reflected right
back would give this line. And then the top and bottom
reflected rays would be traced backward like this. These rays do cross, which means
there is a focal point. This means choice (C) is our
answer. The reflected rays will not be
focused at a point, but the mirror will still have a focal point. It’s the point we’ve identified in
orange.
Let’s now finish this lesson by
reviewing a few key points. In this video, we learned about
convex spherical mirrors. These are convex because the center
of the mirror is closer to an observer than the edges. And they are spherical because the
surface of the mirror is part of the surface of a sphere. The center of that sphere is called
the center of curvature, and the distance between the center of curvature and the
center of the surface of the mirror is called the radius of curvature. When parallel rays of light are
incident on the mirror, those rays reflect back and do not cross. But if we trace the reflected rays
backward, those trace lines do intersect; they meet at the focal point of the
mirror. The distance between the focal
point and the center of the surface of the mirror is called the focal length. The focal length is equal to
one-half the radius of curvature of the mirror. This is a summary of convex
mirrors.